Showing papers on "Spherical shell published in 2005"

Growth and instability in elastic tissues

Abstract: The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.

Steady zonal flows in spherical shell dynamos

Abstract: Convective dynamos in a rotating spherical shell feature steady zonal flows. This process is studied numerically for Prandtl numbers of 0.1 and 1, Ekman numbers in the range $E=\te{-4}$ – $\te{-5}$ , magnetic Prandtl number from 0.5 to 10 and Rayleigh numbers up to 100 times supercritical. The zonal flow is mainly of thermal wind origin, and minimizes the shear of the axisymmetric poloidal magnetic field lines, according to Ferraro's law of corotation. The dissipation in the interior of the fluid is mainly ohmic, while the introduction of rigid velocity boundary conditions confines viscous dissipation in the Ekman boundary layers. The root-mean-square amplitude $U_\varphi$ of the zonal flow in the spherical shell scales as $U_\varphi=(F/\Omega)^{0.5}$ , F being the buoyancy flux through the shell and $\Omega$ the rotation rate. As a consequence of the corotation law, this scaling relationship is remarkably independent of the magnetic field amplitude. It does not depend on thermal, kinematic and magnetic diffusivities, owing to the large-scale and steady nature of forcing and dissipative processes. The scaling law is in agreement with the zonal-flow amplitude at the external boundary of the Earth's liquid core.

Newtonian fluid meets an elastic solid: coupling lattice Boltzmann and lattice-spring models.

Abstract: We integrate the lattice Boltzmann model (LBM) and lattice spring model (LSM) to capture the coupling between a compliant bounding surface and the hydrodynamic response of an enclosed fluid. We focus on an elastic, spherical shell filled with a Newtonian fluid where no-slip boundary conditions induce the interaction. We calculate the "breathing mode" oscillations for this system and find good agreement with analytical solutions. Furthermore, we simulate the impact of the fluid-filled, elastic shell on a hard wall and on an adhesive surface. Understanding the dynamics of fluid-filled shells, especially near adhesive surfaces, can be particularly important in the design of microcapsules for pharmaceutical and other technological applications. Our studies reveal that the binding of these capsules to specific surfaces can be sensitive to the physical properties of both the outer shell and the enclosed fluid. The integrated LBM-LSM methodology opens up the possibility of accurately and efficiently capturing the dynamic coupling between fluid flow and a compliant bounding surface in a broad variety of systems.

Two-Step Approach to Prediction of Asphalt Concrete Modulus from Two-Phase Micromechanical Models

Abstract: A two-step approach is proposed to predict the modulus of asphalt concrete from existing micromechanical models. The asphalt concrete microstructure is represented by a two-phase model, which consists of a large spherical aggregate particle surrounded by a spherical shell of fine aggregate-filler-binder mixture as the matrix. The fine aggregate-filler-binder mixture is further represented by a two-phase model, which treats fine aggregate as a spherical inclusion and the mixture of filler and binder as the matrix. The modulus of asphalt concrete is predicted from the volumetric fractions, Poisson’s ratios, and moduli of the aggregate and filler-binder mixture by applying the appropriate two-phase models in two steps. An asphalt concrete and two mixtures that replicate the fine aggregate-filler-binder submixture and the filler-binder submixture in the asphalt concrete have been tested for modulus. The tests results show that the predicted results from the appropriate models reasonably approximate the measur...

Global Three-dimensional Flow of a Neutron Superfluid in a Spherical Shell in a Neutron Star

Abstract: We integrate for the first time the hydrodynamic Hall-Vinen-Bekarevich-Khalatnikov equations of motion of a 1S0-paired neutron superfluid in a rotating spherical shell, using a pseudo-spectral collocation algorithm coupled with a time-split fractional scheme. Numerical instabilities are smoothed by spectral filtering. Three numerical experiments are conducted, with the following results. (1) When the inner and outer spheres are put into steady differential rotation, the viscous torque exerted on the spheres oscillates quasi-periodically and persistently (after an initial transient). The fractional oscillation amplitude (~10-2) increases with the angular shear and decreases with the gap width. (2) When the outer sphere is accelerated impulsively after an interval of steady differential rotation, the torque increases suddenly, relaxes exponentially, then oscillates persistently as in (1). The relaxation timescale is determined principally by the angular velocity jump, whereas the oscillation amplitude is determined principally by the gap width. (3) When the mutual friction force changes suddenly from Hall-Vinen to Gorter-Mellink form, as happens when a rectilinear array of quantized Feynman-Onsager vortices is destabilized by a counterflow to form a reconnecting vortex tangle, the relaxation timescale is reduced by a factor of ~3 compared to (2), and the system reaches a stationary state in which the torque oscillates with fractional amplitude ~10-3 about a constant mean value. Preliminary scalings are computed for observable quantities such as angular velocity and acceleration as functions of the Reynolds number, angular shear, and gap width. The results are applied to the timing irregularities (e.g., glitches and timing noise) observed in radio pulsars.

A finite-volume solution method for thermal convection and dynamo problems in spherical shells

Abstract: SUMMARY We present a novel application of a finite-volume technique to the numerical simulation of thermal convection within a rapidly rotating spherical shell. The performance of the method is extensively tested against a known standard solution at moderate Ekman number. Models at lower Ekman number demonstrate the potential of the method in a parameter range more appropriate to the flow in the molten metallic core of planetary interiors. In addition we present results for the magnetohydrodynamic dynamo problem. In order to avoid the need to solve for the magnetic field in the exterior, we use an approximate magnetic boundary condition. Compared with the geophysically relevant case of insulating boundaries, it is shown that the qualitative structures of the flow and the magnetic field are similar. However, a more quantitative comparison indicates that mean flow velocity and mean magnetic field strength are affected by the boundary conditions by about 20 per cent.

Donaldson theory on non-Kählerian surfaces and class VII surfaces with b2=1

Abstract: We prove that any class VII surface with b2=1 has curves. This implies the “Global Spherical Shell conjecture” in the case b2=1:

Buckling of spherical shells adhering onto a rigid substrate.

Abstract: Deformation of a spherical shell adhering onto a rigid substrate due to van der Waals attractive interaction is investigated by means of numerical minimization (conjugate gradient method) of the sum of the elastic and adhesion energies. The conformation of the deformed shell is governed by two dimensionless parameters, i.e., Cs/epsilon and Cb/epsilon where Cs and Cb are respectively the stretching and the bending constants, and epsilon is the depth of the van der Waals potential between the shell and substrate. Four different regimes of deformation are characterized as these parameters are systematically varied: (i) small deformation regime, (ii) disk formation regime, (iii) isotropic buckling regime, and (iv) anisotropic buckling regime. By measuring the various quantities of the deformed shells, we find that both discontinuous and continuous bucking transitions occur for large and small Cs/epsilon, respectively. This behavior of the buckling transition is analogous to van der Waals liquids or gels, and we have numerically determined the associated critical point. Scaling arguments are employed to explain the adhesion induced buckling transition, i.e., from the disk formation regime to the isotropic buckling regime. We show that the buckling transition takes place when the indentation length exceeds the effective shell thickness which is determined from the elastic constants. This prediction is in good agreement with our numerical results. Moreover, the ratio between the indentation length and its thickness at the transition point provides a constant number (2–3) independent of the shell size. This universal number is observed in various experimental systems ranging from nanoscale to macroscale. In particular, our results agree well with the recent compression experiment using microcapsules.

Multiple axis gimbal employing nested spherical shells

Abstract: A multi-axis gimbal has each axis defined by a respective spherical shell driven by a flat, compact motor attached to the driven shell and to a next outer shell (or to an external mounting platform, in the case of the outermost shell). The shells rotate about respective axes. In one configuration, the outermost shell is referred to as the “azimuth” shell because in use it rotates about a vertical axis. The next inner shell is an elevation shell that rotates about a first horizontal axis that is orthogonal to the axis of the camera or other sensor payload. An optional third shell can be used to provide “roll” motion, such as rotating a camera about its axis to obtain a particular rotational orientation with respect to a target.

Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell

Abstract: Frequency shifts, radiative decay rates, the Ohmic loss contribution to the nonradiative decay rates, fluorescence yield, and photobleaching of a two-level atom radiating anywhere inside or outside a complex spherical nanoshell, i.e., a stratified sphere consisting of alternating silica and gold concentric spherical shells, are studied. The changes in the spectroscopic properties of an atom interacting with complex nanoshells are significantly enhanced, often more than two orders of magnitude, compared to the same atom interacting with a homogeneous dielectric sphere. The detected fluorescence intensity can be enhanced by 5 or more orders of magnitude. The changes strongly depend on the nanoshell parameters and the atom position. When an atom approaches a metal shell, decay rates are strongly enhanced yet fluorescence yield exhibits a well-known quenching. Rather contra-intuitively, the Ohmic loss contribution to the nonradiative decay rates for an atomic dipole within the silica core of larger nanoshells may be decreasing when the silica core–inner gold shell interface is approached. The quasi-static result that the radial frequency shift in a close proximity of a spherical shell interface is approximately twice as large as the tangential frequency shift appears to apply also for complex nanoshells. Significantly modified spectroscopic properties (see computer program available at http://www.wave-scattering.com ) can be observed in a broad band comprising all (nonresonant) optical and near-infrared wavelengths.

Stokes flow inside a porous spherical shell: Stress jump boundary condition

Abstract: An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkman’s equation for the flow in the porous region is discussed At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used The drag and torque are found by deriving the corresponding Faxen’s laws It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient Critical permeability is found for which drag and torque change their behavior As a limiting case the corresponding Faxen’s laws for the rigid spherical shell with internal singularities has been obtained

Rayleigh-Benard Convection in Spherical Shell with Infinite Prandtl Number at High Rayleigh Number

Abstract: Simulations of the Rayleigh-Benard convection with infinite Prandtl number ( Pr) and high Rayleigh numbers (Ra) in the spherical shell geometry are carried out to understand the thermal structure of the mantle and the evolution of the Earth. We focus on the features of the convection with the most basic set- ting, so the viscosity is assumed to be constant and other complexities of the mantle are not introduced. We have succeeded in calculating the thermal convection in the spherical shell with Ra up to 10 8 , and attained the numerical results for Ra ranging five orders above the critical value. For all Ra, the convection pattern is illustrated as follows; the sheet-shaped downwelling and upwelling flows originate from the boundary layers and concentrate gradually into cylindrical flows. We have examined the relationship between Ra and the Nusselt number (Nu), and obtained that Nu is proportional to Ra 0.30 . The exponent is larger than those of the existing studies. In addition, we quantify the convection pattern by the power spectrum of the temperature field for each depth in terms of spherical harmonic degrees. The analysis reveals that the structural scale of convection differs between the boundary region and the isothermal core region. The structure near the bound- ary region is characterized by the cell type structure constructed by the sheet-shaped downwelling and upwelling flows, and that of the core region by the plume type structure which consists of the cylindrical flows.

Improved Finite Element Analysis of Multilayered, Doubly Curved Composite Shells:

Abstract: An improved finite element model for the bending and free vibration analysis of doubly curved, laminated composite shells having spherical and ellipsoidal shapes is presented. The present formulation is based on the stress resultant-type Koiter’s shell theory and no restriction is imposed on the magnitude of curvature components to capture the deep and shallow shell cases. The twist curvature component is incorporated along with the normal curvatures to keep the strain equations complete. Transverse shear deformation is also considered according to Mindlin’s hypotheses. Both the Lagrangian and Serendipity families of isoparametric elements are incorporated in the analysis code to study their performance in different problems. The present five-DOF shell formulation is kept sufficiently general to capture both the bending- and membrane-dominated problems. The accuracy and efficiency of the proposed finite element are illustrated by examples and are compared with those existing in the open literature. The co...

Dynamo action in a rotating spherical shell at high Rayleigh numbers

Abstract: We investigate convection-driven dynamos in a rotating spherical shell with the Rayleigh number Ra up to about 53 times the critical value Rac, emphasizing Rayleigh number dependence of the thermal convection and the magnetic field generated by dynamo action. The Rayleigh numbers used in calculations are chosen so as to be in a range which allows us to study the sequence of bifurcation. In the low-Ra-dynamo regime, the flow structure is characterized by columnar convection cells, which mainly generate the magnetic field that is predominantly dipolar. Force balance is essentially in a geostrophic state. Both the magnetic energy and the kinetic energy increase with increase in Ra. In the moderate-Ra-dynamo regime, convective motions appear inside the tangent cylinder (TC), where the azimuthal magnetic field is generated through the so-called ω effect. However, the magnetic energy shows saturation due to relatively inefficient magnetic field generation. In the high-Ra-dynamo regime, dominance of convection i...

Free Vibration of Sandwich Laminates with Two Higher-order Shear Deformable Facet Shell Element Models

Abstract: Two higher-order shear deformable finite element models using a higher-order facet shell element are presented for the free vibration analysis of layered anisotropic sandwich laminates. One of the ...

Multiplicity of nonlinear thermal convection in a spherical shell.

Abstract: Linear and weakly nonlinear thermal convection in a moderately thin spherical shell in the presence of a spherically symmetric gravity subject to a spherically symmetric boundary condition is systematically investigated through fully three-dimensional numerical simulations. The convection problem is self-adjoint and the linear convective stability is characterized by l, the degree of a spherical harmonics Yml (theta,phi). While the radial structure of the linear convection is determined by the stability analysis, there exists a (2l + 1)-fold degeneracy in the horizontal structure of the spherical convection. When l = O(10) , i.e., in a moderately thin spherical shell, the removal or partial removal of the degeneracy represents a mathematically difficult, physically not well-understood problem. By starting with carefully chosen initial conditions, we are able to obtain a variety of nonlinear convective flows at exactly the same parameters near the onset of convection, including steady axisymmetric convection, steady azimuthally periodic convection, steady azimuthally nonperiodic convection, equatorially asymmetric convection, and steady convection in the form of a single giant spiral roll covering the whole spherical shell which is stable and robust for a wide range of the Prandtl number.

Contact Interaction of Crack Lips in Shallow Shells in Bending with Tension

Abstract: In the two-dimensional statement, we consider a problem of bending and tension of a shallow cracked shell. The effect of crack closure caused by bending is described within the framework of the classical theory of shells according to the model of contact of crack lips along a line in the face surface. We study the influence of contact interaction of crack lips and surface curvature on the stressed state and limiting equilibrium of cylindrical and spherical shells.